Non-parametric adaptive estimation of the drift for a jump diffusion process

In this article, we consider a jump diffusion process ( X t ) t ≥ 0 observed at discrete times t = 0 , Δ , … , n Δ . The sampling interval Δ tends to 0 and n Δ tends to infinity. We assume that ( X t ) t ≥ 0 is ergodic, strictly stationary and exponentially β -mixing. We use a penalised least-square approach to compute two adaptive estimators of the drift function b . We provide bounds for the risks of the two estimators.